1. Field of the Invention
In general, the field of the invention concerns tomographic reconstruction and more particularly the reconstruction of a sequence of 3D image(s) describing a moving object to be imaged.
2. Description of Related Art
Tomography allows images of an object to be obtained in the form of slices of a region of interest, by acquiring projections. FIG. 1 schematically illustrates the acquisition of 2D images of an object such as an organ, and the tomographic reconstruction of a 3D image of this organ. The reconstruction of a 3D image by tomography comprises the emission of X-rays 10 from a source towards the organ 12, the X-rays being emitted at different angles lε{1, . . . , L} which define the trajectory Tr of the source (commonly an angle of rotation). After passing through the organ 12, the X-rays 10 are detected by a detector (not illustrated) to form a set of 2D projections. There are as many 2D projections acquired as angles under consideration (i.e. L acquisitions). Acquisition is initiated by the detector, which is located facing the X-ray source 11, e.g. a digital camera.
One application of tomography is the detection and characterization of a body lesion e.g. stenosis of a patient's vessel. The acquired 2D projections are used to reconstruct a 3D image of the object. This 3D image is more precisely a 3D mapping of the X-ray attenuation coefficients of the exposed medium. It is by means of this mapping that a radiologist practitioner interprets this image in relation to observed differences in contrast.
An iterative 3D reconstruction method is known. This method is based on a discrete matrix expression of the tomographic reconstruction problem. The method is implemented in a processing unit of a medical imaging system. More precisely, the problem is modelled by the following equation:Rf=p where p is the vector of the L acquired projections, R is a projection operator which models the tomographic imaging system and its trajectory for acquisition of the L projections p, and f is the 3D image of the object to be reconstructed. The tomographic reconstruction problem is to determine f having knowledge of p and R.
One known solution to the above-mentioned equation is the solving of the following criterion:
         {                                                      Q              ⁡                              (                g                )                                      =                                          1                2                            ⁢                                                                                      Rg                    -                    p                                                                    2                2                                                                                                    g              *                        =                                                            arg                  ⁢                                                                          ⁢                  min                                g                            ⁢                              Q                ⁡                                  (                  g                  )                                                                        where ∥ ∥2 symbolizes the Euclidian norm called L2. Minimization of the criterion Q(g) relative to g gives good results (g*≈f) when the set of projections p is such that L is large (typically several hundred) and when the set of L angles covers at least 180°. These conditions are commonly verified when the organ is static during the time needed to perform a rotation of the imaging system.
A problem arises when the organ is in motion and is therefore represented by a series of 3D images {right arrow over (f)}={f(t1), . . . , f(tN)} where f(tn) represents the object at the position referenced tn. The set of projections p is interpreted as a sequence of sets of projections {right arrow over (p)}={p(t1), . . . , p(tN)} where each p(tn) contains the angles available when the object is in the position referenced tn. With a single rotation of the system, the reconstruction of f(tn) is degraded.
if the motion is ignored, and reconstruction is performed by minimizing the criterion Q(g), the solution obtained is:
      g    *    =                              arg          ⁢                                          ⁢          min                g            ⁢              Q        ⁡                  (          g          )                      =                  ∑                  n          =          1                N            ⁢              f        ⁡                  (                      t            n                    )                    It is degraded by object motion which is averaged on the reconstruction. The modelling of the trajectory of the system reduced to the angles of the projections p(tn) is denoted R(tn). The 3D image reconstructed from:
         {                                                      Q              ⁡                              (                                  g                  ,                                      t                    n                                                  )                                      =                                          1                2                            ⁢                                                                                                                                    R                        ⁡                                                  (                                                      t                            n                                                    )                                                                    ⁢                      g                                        -                                          p                      ⁡                                              (                                                  t                          n                                                )                                                                                                              2                2                                                                                                                    g                *                            ⁡                              (                                  t                  n                                )                                      =                                                            arg                  ⁢                                                                          ⁢                  min                                g                            ⁢                              Q                ⁡                                  (                                      g                    ,                                          t                      n                                                        )                                                                        is degraded since R(tn) differs from R in that the number of projections of R(tn), lower than L, is too small and in that the angle coverage is potentially less than 180° C. The sampling condition therefore requires that each p(tn) contains L projections and covers 180° C. It can then be seen that {right arrow over (p)} must contain N×L projections. Yet, such acquisition cannot be envisaged since it implies multiplying N times the X-ray dose (and optionally the quantity of contrast product) and additionally it lasts N times longer.
One alternative consists of integrating the motion of the object by means of a sequence of operators M={M(t1), . . . , M(tN)} which exactly models object motion i.e. such that:∀n,M(tn)f(tn)=f(tref)where f(tref) is the 3D image of the object at a reference position. Reconstruction is then obtained by minimizing the criterion:
         {                                                      Q              ⁡                              (                                  g                  ,                  M                  ,                                      t                    n                                                  )                                      =                                          1                2                            ⁢                                                                                                                                    R                        ⁡                                                  (                                                      t                            n                                                    )                                                                    ⁢                                                                        M                                                      -                            1                                                                          ⁡                                                  (                                                      t                            n                                                    )                                                                    ⁢                      g                                        -                                          p                      ⁡                                              (                                                  t                          n                                                )                                                                                                              2                2                                                                                                                    g                *                            ⁡                              (                                  t                  ref                                )                                      =                                                            arg                  ⁢                                                                          ⁢                  min                                g                            ⁢                                                ∑                                      n                    =                    1                                    N                                ⁢                                  Q                  ⁡                                      (                                          g                      ,                      M                      ,                                              t                        n                                                              )                                                                                          so that the complete sequence {right arrow over (p)}={p(t1), . . . , p(tN)} is used to obtain a single image g*(tref) from which each position g*(tn)=M−1(tn)g*(tref) can be inferred. However, knowledge of M is most often only approximate or only valid in a restricted part of the image. For example, in the document [Blondel C, Malandain G, Vaillant R, Ayache N., “Reconstruction of coronary arteries from a single rotational X-ray projection sequence” IEEE Trans. Med. Imaging 25(5):653-63], the motion of the coronaries is estimated in the projected images and integrated in the reconstruction. However, only the image of the coronaries is obtained, all the other structures being removed from the projective images by pre-processing.
Alternatively, reconstruction is obtained independently for each position tn by minimizing the criterion:
         {                                                      S              ⁡                              (                g                )                                      =                                                                                                W                    xyz                                    ⁢                  g                                                            1                                                                                                      g                *                            ⁡                              (                                                      t                    n                                    ,                  λ                                )                                      =                                                            arg                  ⁢                                                                          ⁢                  min                                g                            ⁢                              {                                                      λ                    ⁢                                                                                  ⁢                                          S                      ⁡                                              (                        g                        )                                                                              +                                      Q                    ⁡                                          (                                              g                        ,                                                  t                          n                                                                    )                                                                      }                                                        where ∥ ∥1 is the so-called L1 norm, Wxyz is a spatial transformation enabling the individual compression of the 3D images (e.g. wavelets, gradient, identity (denoted Id . . . ) and λ is a prior fixed scalar. These compressed images have not yet given any convincing results for clinical application. It is on reconstructing solely the coronary vessels, after removing all other information from the projections by pre-processing [Hansis E, Carrol Schäfer D, Dössel O, Grass M., “High Quality 3-D coronary artery imaging on an interventional C-arm X-ray system” Med Phys. 37(4):1601-9] that the criterion Wxyz=Id is known to produce a sparse image of the coronaries i.e. equal to 0 in all the pixels except the vessels whose volume is known to be very restricted (a few percent in the 3D image).
In the document [Chen G H, Tang J, Leng S., “Prior image constrained compressed sensing (PICCS): a method to accurately reconstruct dynamic CT images from highly undersampled projection data sets” Med. Phys. 35(2): 660-3] it is shown that it is possible to avoid the pre-processing of projection images, provided a strong assumption is made of prior image knowledge denoted fp, close to the solution. The constraint S(g) is then replaced by αS(g)+(1−α)S(g−fp) where 0≦α<1. Variants of the cited techniques are compared in the document [Bergner F, Berkus T, Oelhafe M, Kunz P, Pan T, Grimmer R, Ritschl L, Kachelrieβ M, <<An investigation of 4D cone-beam CT algorithms for slowly rotating scanners>> Med. Phys. 37(9): 5044-53].